The rise of Euclidean axiomatics

Colin Maclarty

Case Western Reserve University, Cleveland

Abstract: 
We apply ideas from Ian Mueller and David Fowler to the rise of Euclidean axiomatics. Hippocrates of Chios invokes a curvilinear case of the Pythagorean theorem about 200 years before Euclid. A Euclid-style treatment of this case would require extending proofs in Euclid's Books I, VI, and XII. The evidence suggests Hippocrates used an argument in a few lines, while Euclid I.47 takes scores of pages just for the familiar case of squares on a right triangle. Euclid's radically longer and radically more explicit Elements eclipsed all earlier geometries as utterly as Homer eclipsed all earlier epic poets, and became a paradigm framework for mathematics to this day.